Algorithms and Economics of Networks

Algorithms and Economics of Networks

Abraham Flaxman and Vahab Mirrokni, Microsoft Research

Topics

Algorithms for Complex Networks Economics and Game Theory

Algorithms for Large Networks TraceRoute Sampling

Where do networks come from? Network Formation Link Analysis and Ranking

What Can Link Structure Tell Us About Content? Hub/Authority and Page-Rank Algorihtms

Clustering  Inferring Communities from Link Structure Local Partitioning Based on Random Walks Spectral Clustering Balanced Partitioning.

Diffusion and Contagion in Networks Spread of Influence in Social Networks.

Rank Aggregation    Recent Algorithmic Achievements.

Logistics Course Web Page:

http://www.cs.washington.edu/education/courses/cse599m/07sp/

Course Work

Scribe One Topic One Problem Set due Mid-May One Project

Contact: {Abie,Mirrokni}@Microsoft.com

Why do we study game theory?

Selfish Agents Many networking systems consist of

self-interested or selfish agents. Selfish agents optimize their own

objective function. Goal of Mechanism Design:

encourage selfish agents to act socially. Design rewarding rules such that when

agents optimize their own objective, a social objective is met.

Self-interested Agents How do we study these systems? Model the networking system as a

game, and Analyze equilibrium points. Compare the social value of

equilbirim points to global optimum.

Algorithmic Game Theory Important Factors:

Existence of equilibria as as subject of study.

Performance of the output (Approximation Factor).

Convergence (Running time) Computer Science

Economics of Networks Lack of coordination in networks    

Equilibrium Concepts: Strategic Games and Nash equilibria Price of Anarchy.

Load Balancing Games. Selfish Routing Games and Congestion Games. Distributed Caching and Market Games. Efficiency Loss in Bandwidth Allocation Games.

Coordination Mechanisms Local Algorithmic Choices Influence the Price of Anarchy.

Market Equilibria and Power Assignment in Wireless Networks. Algorithms for Market Equilibria. Power Assignment for Distributed Load Balancing in Wireless

Networks.  Convergence and Sink Equilibria

Best-Response dynamics in Potential games. Sink Equilibria : Outcome of the Best-response Dynamics. Best response Dynamics in Stable Matchings.

Basics of Game Theory

Game Theory Was first developed to explain the

optimal strategy in two-person interactions

Initiated for Zero-Sum Games, and two-person games.

We study games with many players in a network.

Example: Big Monkey and Little Monkey [Example by Chris Brook, USFCA] Monkeys usually eat ground-level fruit Occasionally climb a tree to get a

coconut (1 per tree) A Coconut yields 10 Calories Big Monkey spends 2 Calories climbing

the tree. Little Monkey spends 0 Calories

climbing the tree.

Example: Big Monkey and Little Monkey If BM climbs the tree

BM gets 6 C, LM gets 4 C LM eats some before BM gets down

If LM climbs the tree BM gets 9 C, LM gets 1 C BM eats almost all before LM gets down

If both climb the tree BM gets 7 C, LM gets 3 C BM hogs coconut

How should the monkeys each act so as to maximize their own calorie gain?

Example: Big Monkey and Little Monkey Assume BM decides first

Two choices: wait or climb LM has also has two choices after BM

moves. These choices are called actions

A sequence of actions is called a strategy.

Example: Big Monkey and Little Monkey

Big monkey w

w w

c

cc

0,0

Little monkey

9,1 6-2,4 7-2,3

What should Big Monkey do?• If BM waits, LM will climb – BM gets 9• If BM climbs, LM will wait – BM gets 4• BM should wait.• What about LM?• Opposite of BM (even though we’ll never get to the right side of the tree)

Example: Big Monkey and Little Monkey These strategies (w and cw) are called

best responses. Given what the other guy is doing, this is the

best thing to do. A solution where everyone is playing a

best response is called a Nash equilibrium. No one can unilaterally change and improve

things. This representation of a game is called

extensive form.

Example: Big Monkey and Little Monkey What if the monkeys have to decide

simultaneously? It can often be easier to analyze a

game through a different representation, called normal form

Strategic Games: One-Shot Normal-Form Games with Complete Information…

Normal Form Games Normal form game (or Strategic games)

finite set of players {1, …, n} for each player i, a finite set of actions (also

called pure strategies): si1, …, si

k

strategy profile: a vector of strategies (one for each player)

for each strategy profile s, a payoff Pis to

each player

Example: Big Monkey and Little Monkey This Game has two Pure Nash equilibria A Mixed Nash equilibrium: Each Monkey

Plays each action with probability 0.5

Big Monkey

Little Monkey

c

c w

w

5,3 4,4

0,09,1

Nash’s Theorem Nash defined the concept of mixed

Nash equilibria in games, and proved that:

Any Strategic Game possess a mixed Nash equilibrium.

Best-Response Dynamics State Graph: Vertices are strategy profiles. An

edge with label j correspond to a strict improvement move of one player j.

Pure Nash equilibria are vertices with no outgoing edge.

Best-Response Graph: Vertices are strategy profiles. An edge with label j correspond to a best-response of one player j.

Potential Games: There is no cycle of strict

improvement moves There is a potential function for the game.

BM-LM is a potential game. Matching Penny game is not.

Example: Prisoner’s Dilemma Defect-Defect is the only Nash

equilibrium. It is very bad socially.

cooperate defect

defect 10,0

0,10

1,1

5,5

Row

Column

cooperate

Price of Anarchy The worst ratio between the social

value of a Nash equilibrium and social value of the global optimal solution.

An example of social objective: the sum of the payoffs of players.

Example: In BM-LM Game, the price of anarchy for pure NE is 8/10. POA for mixed NE is 6.5/10.

Example: In Prisoner’s Dilemma, the price of anarchy is 2/10.

Load Balancing Games n players/jobs, each with weight wi

m strategies/machines Outcome M: assignment jobs → machines J( j ): jobs on machine j L( j ) = Σi in J( j ) wi : load of j R( j ) = f j ( L( j ) ): response time of j

f j monotone, ≥ 0 e.g., f j (L)=L / s j

(s j is the speed of machine j) NE: no job wants to switch, i.e., for any i in J( j ) f j ( L( j ) ) ≤ f k ( L( k ) + w j ) for all k ≠ j

32

2

4

m1 m2

Load Balancing Games(parts of slides from E.

Elkind, warwick) n players/jobs, each with weight wi m strategies/machines Outcome M: assignment jobs → machines J( j ): jobs on machine j L( j ) = Σi in J( j ) wi : load of j R( j ) = f j ( L( j ) ): response time of j

f j monotone, ≥ 0 e.g., f j (L)=L / s j (s j is the speed of machine j)

NE: no job wants to switch, i.e., for any i in J( j ) f j ( L( j ) ) ≤ f k ( L( k ) + w j ) for all k ≠ j

Social Objective: worst response time maxj R(j)

32

2

4

m1 m2

Load Balancing Games Theorem: if all response times are

nonegative increasing functions of the load, pure NE exists.

Proof: start with any assignment M order machines by their response times allow selfish improvement; reorder each assignment is lexicographically better

than the previous one jobs migrate from left to right

Load Balancing Games: POA Social Objective: worst response time maxj R(j) Theorem: if fj(L) = L (response time = load),

Worst Pure Nash/Opt ≤ 2. Proof:

M: arbitrary pure Nash, M’: Opt j: worst machine in M, i.e., C( M )=RM( j ) k: worst machine in M’, i.e., C( M’ )=RM’( k )

there is an l s.t. RM( l ) ≤ RM’( k ) (averaging argument) w = max wi ; RM’( k ) ≥ w RM( j ) - RM( l ) ≥ 2RM’( k ) - RM’( k ) ≥ w => in M, there is a job that wants to switch from j to l.

C(M) ≥ 2 * C(M’) impliesRM( j ) ≥ 2 * RM’( k )

Price of Anarchy for Load Balancing POA for Mixed Nash Equilibria P||C max : for fj(L) = L, POA is 2-2/m+1. Q||C max : for f j (L)=L / s j, POA is

O(logm/loglogm). R||C max : for fj(L) = L and each job can

be assigned to a subset of machines, POA is O(logm/loglogm).

Will give some proofs in the lecture on coordination mechanisms.

We Know Normal Form Games Pure and Mixed Nash Equilibria Best-Response Dynamics, State Graph Potential Games Price of Anarchy Load Balancing Games

We didn’t talk about Other Equilibrium Concepts: Subgame

Perfect Equilibria, Correlated Equilibria, Cooperative Equilibria

Price of Stability

Next Lecture. Congestion Games

Rosenthal’s Theorem: Congestion games are potential Games:

Market Sharing Games Submodular Games

Vetta’s Theorem: Price of anarchy is ½ for these games.

Selfish Routing Games